Journal of Modern Dynamics (JMD)

Regularity of conjugacies of algebraic actions of Zariski-dense groups

Pages: 509 - 540, Issue 3, July 2008      doi:10.3934/jmd.2008.2.509

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Alexander Gorodnik - School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom (email)
Theron Hitchman - Department of Mathematics, University of Northern Iowa, Cedar Falls, IA 50614-0506, United States (email)
Ralf Spatzier - Department of Mathematics, 2074 East Hall, 530 Church Street, University of Michigan, Ann Arbor, MI 48109-1043, United States (email)

Abstract: Let $\alpha_0$ be an affine action of a discrete group $\Gamma$ on a compact homogeneous space $X$ and $\alpha_1$ a smooth action of $\Gamma$ on $X$ which is $C^1$-close to $\alpha_0$. We show that under some conditions, every topological conjugacy between $\alpha_0$ and $\alpha_1$ is smooth. In particular, our results apply to Zariski-dense subgroups of $SL_d(\mathbb{Z})$ acting on the torus $\mathbb{T}^d$ and Zariski-dense subgroups of a simple noncompact Lie group $G$ acting on a compact homogeneous space $X$ of $G$ with an invariant measure.

Keywords:  Group actions, conjugacies, rigidityh.
Mathematics Subject Classification:  Primary: 37C85, Secondary: 53C24.

Received: April 2008;      Available Online: April 2008.